For let A by multiplying itself make D;
therefore D is square.
Now, since A by multiplying itself has made D, and by multiplying B has made C,
therefore, as A is to B, so is D to C. [VII. 17]
And, since D is square, and C is so also,
therefore D, C are similar plane numbers.
Therefore one mean proportional number falls between D, C. [VIII. 18]
And, as D is to C, so is A to B;
therefore one mean proportional number falls between A, B also. [VIII. 8]
But, if one mean proportional number fall between two numbers, they are similar plane numbers; [VIII. 20]
therefore A, B are similar plane numbers.
Q. E. D.
Complete text
Book I